Question

​(Bond valuation​) You are examining three bonds with a par value of ​$1 comma 000 ​(you receive ​$1 comma 000 at​ maturity) and are concerned with what would happen to their market value if interest rates​ (or the market discount​ rate) changed. The three bonds are Bond Along dasha bond with 5 years left to maturity that has an annual coupon interest rate of 8 ​percent, but the interest is paid semiannually. Bond Blong dasha bond with 10 years left to maturity that has an annual coupon interest rate of 8 ​percent, but the interest is paid semiannually. Bond Clong dasha bond with 15 years left to maturity that has an annual coupon interest rate of 8 ​percent, but the interest is paid semiannually. What would be the value of these bonds if the market discount rate were a. 8 percent per year compounded​ semiannually? b. 5 percent per year compounded​ semiannually? c. 15 percent per year compounded​ semiannually? d. What observations can you make about these​

Answer 1

Bond A, 5 years to maturity, semiannual coupons, 8%

Bond B, 10 years to maturity, annual coupon, 8%

Bond C, 15 years to maturity, semiannual coupon, 8%

a) market rate 8% semiannual

Bonds A and C will be worth $1,000 (par value)

price of bond B:

• effective interest rate = 1.04² - 1 = 8.16%
• PV of face value = $1,000 / 1.04²⁰ = $456.39
• PV of coupon payments = $80 x 6.66192 (PV ordinary annuity factor, 8.16%, 10 periods) = $532.95

market price = $989.34

b) price of bond A:

PV of face value = $1,000 / 1.025¹⁰ = $781.98

PV of coupon payments = $40 x 8.75206 (PV ordinary annuity factor, 2.5%, 10 periods) = $350.08

market price = $1,132.06

price of bond B:

• effective interest rate = 1.025² - 1 = 5.0625%
• PV of face value = $1,000 / 1.025²⁰ = $610.27
• PV of coupon payments = $80 x 7.69817 (PV ordinary annuity factor, 5.0625%, 10 periods) = $615.85

market price = $1,226.12

price of bond C:

PV of face value = $1,000 / 1.025³⁰ = $476.74

PV of coupon payments = $40 x 20.93029 (PV ordinary annuity factor, 2.5%, 30 periods) = $837.21

market price = $1,313.95

c) price of bond A:

PV of face value = $1,000 / 1.075¹⁰ = $485.19

PV of coupon payments = $40 x 6.86408 (PV ordinary annuity factor, 7.5%, 10 periods) = $274.56

market price = $759.75

price of bond B:

• effective interest rate = 1.075² - 1 = 15.5625%
• PV of face value = $1,000 / 1.075²⁰ = $235.41
• PV of coupon payments = $80 x 4.91292 (PV ordinary annuity factor, 15.5625%, 10 periods) = $393.03

market price = $628.44

price of bond C:

PV of face value = $1,000 / 1.075³⁰ = $114.22

PV of coupon payments = $40 x 11.81039 (PV ordinary annuity factor, 7.5%, 30 periods) = $472.42

market price = $586.64

d) If the market rate is lower than the coupon rate, then the bonds will sell at a premium. The longer the maturity date, the larger the variations in market price due to different interest rates. E.g. the 15 year bond is more affected than the 5 year bond.